Final answer:
The Mongol invasion of Anatolia weakened the Seljuk Turks' Sultanate of Rum, allowing the Ottomans to establish an independent state through military victories and effective governance strategies like the millet system.
Step-by-step explanation:
The development of an Ottoman state independent of the Seljuk Turks' Sultanate of Rum was primarily facilitated by the Mongol invasion of Anatolia. This invasion weakened the Seljuk state, leading to power vacuums that the Turks, led by Osman I, exploited. As the Seljuks declined, the Ottomans gained military victories and expanded their territory, laying the foundation for their empire. The Ottoman victory at the Battle of Bapheus in 1302 further solidified their independence from the Seljuk Turks' influence. Moreover, the Ottomans employed strategies such as the millet system to manage their multi-ethnic empire effectively, fostering economic development and forming a strong centralized state.